Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,907 questions
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Minimal volume of 4-manifolds
This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...
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Are there some other notions of "curvature" which measure how space curves?
I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization of ...
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What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?
In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...
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Explicit eigenvalues of the Laplacian
Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...
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How to tackle the smooth Poincaré conjecture
The last remaining problem in this whole "everything is a sphere" business, is the smooth Poincaré conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^...
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Surfaces filled densely by a geodesic
Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...
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Advantages of diffeological spaces over general sheaves
I have been playing with/thinking about diffeological spaces a bit recently, and I would like understand something rather crucial before going further. First a little background:
Diffeological spaces ...
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Can every Lie group be realized as the full isometry group of a Riemannian manifold?
Suppose a finite-dimensional Lie group $G$ is given. Does there exist a connected manifold $M$ and a Riemannian metric $g$, such that $G$ is the full isometry group of $(M,g)$?
For example if I try to ...
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Exotic differentiable structures on R^4?
This was going to be a comment to Differentiable structures on R^3, but I thought it would be better asked as a separate question.
So, it's mentioned in the previous question that $\mathbb{R}^4$ has ...
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Manifold of probability measures: connections between two types of metrics
The space of probability measures could be viewed as an infinite-dimensional manifold, equipped with two possible types of metrics — (1) Wasserstein and (2) Fisher-Rao. Metric (1) is connected with ...
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What else is Seiberg-Witten Theory equal to?
In low-dimensional topology there have been a bunch of invariants defined, and Seiberg-Witten Theory seems to make its appearance in [a lot of] them:
1) Heegaard Floer homology = SW Floer homology (...
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Smoothness of distance function in Riemannian Manifolds
Let $(\mathcal{M},g)$ be a $C^{\infty}$-Riemannian manifold. A basic fact is that $g$ endows the manifold $\mathcal{M}$ with a metric space structure, that is, we can define a distance function $d:\...
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Relative De Rham cohomologies
as far as I know, there are two main ways to have a relative version of De Rham Cohomology for a pair (M,N), where M and N are smooth manifolds and N is a closed (as a topological subspace) ...
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Intuition behind the ricci flow
I hope you don't shoot me for this question.
I try to understand among other things the Ricci flow. However I have no idea of the intuition behind the definition. So my questions is:
What is the ...
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Is it possible to improve the Whitney embedding theorem?
Edited to fix the example, as per Zack's suggestion.
Edit 2: So it turns out that when I think 'manifold' I tend to assume the nicest possible object. As I believe is standard, I would like to ...
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Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
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Geometric imagination of differential forms
In order to explain to non-experts what a vector field is, one usually describes an assignment of an arrow to each point of space. And this works quite well also when moving to manifolds, where a ...
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Is "Cartan's magic formula" due to Élie or Henri?
The formula $\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega)$ is sometimes attributed to Henri Cartan (e.g. Peter Petersen; Riemannian Geometry 2nd ed.; p.380) and sometimes to his father Élie (e.g. ...
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How useful is differential geometry and topology to deep learning?
After seeing this article https://www.quantamagazine.org/an-idea-from-physics-helps-ai-see-in-higher-dimensions-20200109/ I wanted to ask myself how useful of an endeavor would it be if one goes ...
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When are the eigenspaces of the Laplacian on a compact homogeneous space irreducible representations?
I was writing up some notes on harmonic analysis and I thought of a question that
I felt I should know the answer to but didn't, and I hope someone here can help me.
Suppose I have a compact ...
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Real manifolds in a theorem prover?
Which of the formal computer proof verification systems (like Lean, Coq, Agda, Idris, Isabelle-HOL, HOL-Light, Mizar etc) have a basic theory of real manifolds? Up to, say, the definition of a smooth ...
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Is S^2 x S^4 a complex manifold?
As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...
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Some questions about scalar curvature
Recall that the scalar curvature of a Riemannian manifold is given by the trace of the Ricci curvature tensor. I will now summarize everything that I know about scalar curvature in three sentences:
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Vector bundles vs principal $G$-bundles
It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres
$$F=\pi^{-1}(x), \ \ \ x\in B $$
over any $x\in B$, are ...
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Kervaire invariant: Why dimension 126 especially difficult?
Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\...
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Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature
A curve in the plane is determined, up to orientation-preserving
Euclidean
motions, by its curvature function, $\kappa(s)$.
Here is one of my favorite examples, from
Alfred Gray's book,
Modern ...
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Can a topological manifold have different tangent bundles?
We know that the tangent bundles of the sphere arising from different smooth structures are equivalent as vector bundles. Is it right in general? I want to know the relationship between the set of ...
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When is a closed differential form harmonic relative to some metric?
Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$.
Question: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is $g$-harmonic,...
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Is there a mathematical book on general relativity that uses exclusively a coordinate free language even in practical computations?
I would also appreciate if it was as far from the physicists formalism as possible, no abstract indices ,etc. Also I don't consider using a basis or tetrads as coordinate free.
The idea is to use ...
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Is there a general theory of "compactification"?
In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
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Why are differential forms called closed and exact?
It seems to me that "exact" relates to exact differential equation. So, why are they called exact?
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$G_2$ and Geometry
In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie ...
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Modern mathematical books on general relativity
I am looking for a mathematical precise introductory book on general relativity. Such a reference request has already been posted in the physics stackexchange here. However, I'm not sure whether some ...
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Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?
The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...
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Why is cotangent more canonical than tangent?
You don't need a metric to define the differential of a function,
and the cotangent bundle carries a canonical one-form.
But you do need a metric to define the gradient, and the
tangent bundle does ...
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Applications of super-mathematics to non-super mathematics
Supergeometry and more broadly supermathematics has been around for few decades. Since its introduction by physicists, there has been an some mathematical interest in them.
Although interesting in its ...
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What is the best way to draw curvature?
This is more of a pedagogical question rather than a strictly mathematical one, but I would like to find good ways to visually depict the notion of curvature. It would be preferable to have pictures ...
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Kähler structure on cotangent bundle?
The total space of cotangent bundle of any manifold $M$ is a symplectic manifold.
Is it true/false/unknown that for any $M$, $T^*M$ has Kähler structure?
Please support your claim with reference or ...
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How should you explain parallel transport to undergraduates?
The title is a bit deceiving, because what I really mean is the parallel transport that corresponds to the Levi–Civita connection.
This is in the vein of many other questions on mathoverflow:
What is ...
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How to see the Phase Space of a Physical System as the Cotangent Bundle
Two things today motivated this question.
First, the professor said that in a lecture Thurston mentioned
Any manifold can be seen as the configuration space of some physical system.
Clearly we ...
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Has anything precise been written about the Fukaya category and Lagrangian skeletons?
At some point in this past year, some Fukaya people I know got very
excited about the Fukaya categories of symplectic manifolds with "Lagrangian skeletons." As I understand it, a
Lagrangian ...
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geometric interpretation of Lie bracket
On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written:
We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to
which the integral ...
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Mathematical uses of string theory
It is widely believed that correctness of string theory as a physical theory will not be decided in the near future. Regardless whether this will turn out to be correct or not, mathematical concepts ...
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Two definitions of Calabi-Yau manifolds
Why is it that the vanishing of the integral first Chern class of a compact Kahler manifold is equivalent to the canonical bundle being trivial? I can see that it implies that the canonical bundle ...
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Does the Pfaffian have a geometric meaning?
While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph:
"
...In a certain sense, this might be considered a very satisfactory generalization of Gauss-Bonnet. ...
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Is there a geometric construction of hyperbolic Kac-Moody groups?
Just as the theory of finite-dimensional simple Lie algebras is connected to differential geometry and physics via the theory of simple Lie groups, the theory of affine Lie algebras was connected to ...
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Strong Whitney embedding theorem for non-compact manifolds
$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong ...
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Functions whose gradient-descent paths are geodesics
Let $f(x,y)$ define a surface $S$
in $\mathbb{R}^3$ with a unique local minimum at $b \in S$.
Suppose gradient descent from any start point $a \in S$
follows a geodesic on $S$ from $a$ to $b$.
(Q1.)
...
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What is the best way to peel fruit?
A mango made me wonder about this. (See also this question, which is in a similar spirit.)
Fix $L >0$ and a smooth body (possibly nonconvex—pears or bananas are fair game!) $B \subset \mathbb{R}^3$...
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Jet bundles and partial differential operators
A geometric way of looking at differential equations
In the literature for the h-principle (for example Gromov's Partial differential relations or Eliashberg and Mishachev's Introduction to the h-...
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